It's a mix of the two because the gyro is canted at angle such that it is sensitive to both roll rate and and yaw rate (note that it's roll rate, not bank angle).
The result is you start to get a turn indication immediately with the bank (a turn and bank has a slight lag because there is no indication until the heading change has started slightly after the bank), and straight yaw indications without bank are reduced from what they would be in a turn and bank.
In an airplane with a regular turn and bank, if you get bumps that make the nose swing back and forth while the wings remain level, the turn indicator needle will swing from side to side because its gyro doesn't know the airplane isn't banked; it just knows that yaw motions are happening. A turn and bank can be a pain to use in bumpy air in a plane that likes to wag from side to side in turbulence. You can't really use the indication until the slip/skid ball is settled down and centered.
A turn coordinator with its canted gyro does this a lot less because it's only partially sensitive to yaw rates, and because it's also sensitive to roll rates, it gives you an indication of banking into a turn immediately upon dropping the wing.
The overall result is that initial indication is from the roll into or out of the turn, and once the change of bank has stopped, the indication is coming from the yaw rate of the turn itself.
Being sensitive to roll and yaw rates of motion, to work well, a turn coordinator's gyro needs an internal damper to dampen its motion. The turn and bank in my old '68 Cessna Cardinal went bad once when the internal damper failed. The plane symbol would dance around constantly, much worse than a turn and bank needle (which has no damping), making the instrument unusable even though the gyro was still running.
